KILLED
proof of input_zOz2LjEEad.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (24) CdtProblem
    (25) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 781 ms]
    (36) CdtProblem
    (37) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1164 ms]
    (38) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs))
   badd(x, Nil) -> x
   goal(x, y) -> badd(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   goal_2

(c) The following functions are completely defined:

   badd_2

Due to the following rules being added:
none

And the following fresh constants: none

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   badd(x', Cons(x, Cons(x1, xs'))) -> badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(x', xs'))) [2]
   badd(x', Cons(x, Nil)) -> badd(Cons(Nil, Nil), x') [2]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   badd(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, x') :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x'
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(x', xs'))) :|: x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z = x', z' = 1 + x + (1 + x1 + xs')
   goal(z, z') -{ 1 }-> badd(x, y) :|: x >= 0, y >= 0, z = x, z' = y


----------------------------------------

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   badd(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, z) :|: z >= 0, z' - 1 >= 0
   badd(z, z') -{ 2 }-> badd(1 + 0 + 0, badd(1 + 0 + 0, badd(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   goal(z, z') -{ 1 }-> badd(z, z') :|: z >= 0, z' >= 0


----------------------------------------

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

----------------------------------------

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   badd(x', Cons(x, xs)) -> badd(Cons(Nil, Nil), badd(x', xs)) [1]
   badd(x, Nil) -> x [1]
   goal(x, y) -> badd(x, y) [1]

The TRS has the following type information:
badd :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   badd(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   badd(z, z') -{ 1 }-> badd(1 + 0 + 0, badd(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   goal(z, z') -{ 1 }-> badd(x, y) :|: x >= 0, y >= 0, z = x, z' = y

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(21) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
   BADD(z0, Nil) -> c1
   GOAL(z0, z1) -> c2(BADD(z0, z1))
K tuples:none
Defined Rule Symbols:   badd_2, goal_2

Defined Pair Symbols:   BADD_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1


----------------------------------------

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c2(BADD(z0, z1))
Removed 1 trailing nodes:
   BADD(z0, Nil) -> c1

----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
   goal(z0, z1) -> badd(z0, z1)
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
K tuples:none
Defined Rule Symbols:   badd_2, goal_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


----------------------------------------

(25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> badd(z0, z1)

----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
S tuples:
   BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


----------------------------------------

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BADD(z0, Cons(z1, z2)) -> c(BADD(Cons(Nil, Nil), badd(z0, z2)), BADD(z0, z2)) by 
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))

----------------------------------------

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))
S tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0), BADD(z0, Nil))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2


----------------------------------------

(29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
S tuples:
   BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2)))
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


----------------------------------------

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace BADD(z0, Cons(x1, Cons(z1, z2))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2))), BADD(z0, Cons(z1, z2))) by 
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))

----------------------------------------

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
S tuples:
   BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_1, c_2


----------------------------------------

(33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace BADD(z0, Cons(x1, Nil)) -> c(BADD(Cons(Nil, Nil), z0)) by 
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))

----------------------------------------

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
K tuples:none
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


----------------------------------------

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
We considered the (Usable) Rules:none
And the Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( badd_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( Cons_2(x_1, x_2) ) =  	[[0], [4]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 4], [0, 0]] 	* 	x_2
>>>

<<<
 M( Nil ) =  	[[0], [0]]
>>>

Tuple symbols: 
<<<
 M( c_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( BADD_2(x_1, x_2) ) =  	[[0], [2]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
K tuples:
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1


----------------------------------------

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
We considered the (Usable) Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
And the Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( badd_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[2, 0], [0, 1]] 	* 	x_1 	+ 	[[0, 0], [0, 1]] 	* 	x_2
>>>

<<<
 M( Cons_2(x_1, x_2) ) =  	[[0], [2]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 4], [0, 1]] 	* 	x_2
>>>

<<<
 M( Nil ) =  	[[0], [0]]
>>>

Tuple symbols: 
<<<
 M( c_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( BADD_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   badd(z0, Cons(z1, z2)) -> badd(Cons(Nil, Nil), badd(z0, z2))
   badd(z0, Nil) -> z0
Tuples:
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
S tuples:
   BADD(Cons(y1, Nil), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Nil)))
K tuples:
   BADD(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Cons(y3, y4)))))
   BADD(Cons(y1, Cons(y2, Nil)), Cons(z1, Nil)) -> c(BADD(Cons(Nil, Nil), Cons(y1, Cons(y2, Nil))))
   BADD(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(Cons(Nil, Nil), badd(z0, z2)))), BADD(z0, Cons(x2, Cons(z1, z2))))
   BADD(z0, Cons(x1, Cons(x2, Nil))) -> c(BADD(Cons(Nil, Nil), badd(Cons(Nil, Nil), z0)), BADD(z0, Cons(x2, Nil)))
Defined Rule Symbols:   badd_2

Defined Pair Symbols:   BADD_2

Compound Symbols:   c_2, c_1